Implicit Differentiation
Implicit Differentiation
Objectives
- Find the derivative of a complicated function by using implicit differentiation.
- Use implicit differentiation to determine the equation of a tangent line.
Summary
In an equation involving \(x\) and \(y\) where portions of the graph can be defined by explicit functions of \(x\text{,}\) we say that \(y\) is an implicit function of \(x\text{.}\) A good example of such a curve is the unit circle.
We use implicit differentiation to differentiate an implicitly defined function. We differentiate both sides of the equation with respect to \(x\text{,}\) treating \(y\) as a function of \(x\) by applying the chain rule. If possible, we subsequently solve for \(\frac{dy}{dx}\) using algebra.
While \(\frac{dy}{dx}\) may now involve both the variables \(x\) and \(y\text{,}\) \(\frac{dy}{dx}\) still gives the slope of the tangent line to the curve. It may be used to decide where the tangent line is horizontal (\(\frac{dy}{dx} = 0\)) or vertical (\(\frac{dy}{dx}\) is undefined), or to find the equation of the tangent line at a particular point on the curve.